Fix \(\varepsilon >0\) ; the proof by truncation of the weak law of large numbers provided e.g. in [1]}, shows the following: let \(\Sigma _{n} = \sum _{k=1}^{n} ((\widehat{X}^{n}_{k})^{2} - s_{n}^{2})\)  and \(\Sigma _{n}^{\prime } = \sum _{k=1}^{n} ((\widehat{X}^{n}_{k})^{2} - s_{n}^{2}) \mathbb {1}_{\lbrace |(\widehat{X}^{n}_{k})^{2} - s_{n}^{2}| \leqslant n \varepsilon ^{3}\rbrace }\) , then

P(|n-E[n']| > n )
  E[|(Xn1)2 - sn2|] + n  P(|(Xn1)2 - sn2| > n 3)
